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A core prediction of the core accretion model for giant planet formation is that the frequency of readily detectable giant planets should increase with both increasing stellar metallicity and with increasing stellar mass:

It’s now well established that the above diagram is zeroth-order correct, but until fairly recently, the conventional wisdom held that there is little evidence for a strong planet-metallicity correlation among the handful M-dwarf stars (for example, Gliese 876) that are known to harbor giant planets. One is then naturally led to speculate that the odd giant planets in a systems like Gliese 876 might be the outcome of gravitational instability rather than core accretion.

The profusion of molecular lines in the atmospheres of M dwarfs make it hard to determine their metallicities using the techniques of spectral synthesis that work well for hotter stars like the Sun. Fortunately, though, the red dwarfs’ legendary stinginess provides another opportunity for assessing metallicity. Red dwarfs are so thrifty, and they evolve so slowly, that every single one that’s ever formed has barely touched its store of hydrogen. With all the fuel gauges pegged to full, a critical parameter’s worth of confusion is removed. Red dwarfs of a particular mass should form a well-defined one-parameter sequence in the Hertzsprung Russell diagram, and that parameter should be metallicity. If one can accurately plot a particular low-mass star on a color-magnitude diagram, then there should exist a unique and high-quality mapping to both the star’s mass and its metallicity. Physically, an increase in metallicity leads to a higher photospheric opacity, which provides an effective layer of insulation for a star. Add metals to a red dwarf and it will move down and to the right in the Hertzsprung Russell diagram.

Because of the nightmarish complexity of red dwarf atmospheres, it’s not easy to find the calibration that allows one to make the transformation between an observed absolute magnitude and color index (e.g. M_K and V-K) to the stellar mass and metallicity. In 2005, however, Xavier Bonfils and his collaborators made a breakthrough by employing a simple should’ve-thought-of-that-myself technique: Binary stars generally stem from a common molecular cloud core, and so the members of a binary pair should thus generally have very similar metallicities. In particular, if you measure the metallicity of an F, G, or K binary companion to an M-dwarf, then you can assume that the M-dwarf has the same metallicity. Do this often enough, and you can infer the lines of constant M-dwarf metallicity on a color-magnitude diagram. With the calibration in place, metallicity determinations for field red dwarfs are simply a matter of reading off the nearest iso-metallicity locus. Here’s the key diagram from the Bonfils et al. paper:

The puzzling outcome of the Bonfils et al metallicity calibration was that the rare giant-planet bearing M-dwarfs such as Gliese 876 and Gliese 849 didn’t appear to be particularly metal rich, and that worked to undermine confidence in the core accretion picture. One would naively expect that a low-mass disk will need all the help it can get in order to build giant planet cores before the gas is gone. If anything, the planet-metallicity correlation should be strongest among the M-dwarfs.

Important recent progress was made last year by John Johnson and Kevin Apps, who published a reevaluation of Bonfil et al’s. isometallicity loci in the color-magnitude diagram. Johnson and Apps point out that application of the Bonfils et al. calibration produces an aggregate of local M-dwarf stars that have a significantly lower average metallicity than that for the local FGK stars. There’s little reason to expect such a dichotomy, which implies that the Bonfils et al. correlation may be systematically underestimating metallicity by roughly a factor of two. No small potatoes!

Johnson and Apps adjusted the calibration to bring the metallicities of the local M dwarfs into line with the metallicities of the local FGK dwarfs. Here’s a slightly adapted version of their key diagram:

With the revised calibration, Gliese 876 turns up with a metallicity twice that of the Sun, and there is excellent evidence that the planet-metallicity correlation holds strongly for the M dwarfs that harbor relatively massive planets. Furthermore, it’s hard to argue with the two recent papers (one, two) from the California Planet Survey which report the detection of relatively massive planets orbiting two nearby M dwarfs, both of which have extremely high metallicities with the revised calibration.

The statistics are still small-number, but there’s a strong hint that the planet-metallicity correlation for Neptune and sub-Neptune mass planets orbiting M-dwarfs is stronger than it appears to be at FGK (where it’s effectively non-existent). Gliese 176, and Gliese 436, for example, are both quite metal-rich. I bet that a survey like Mearth could jack up its yield by shading its telescope visits to favor the high-metallicity stars on the observing list…

Indeed, if we plot Gliese 1214 (V=15.1±0.6, K=8.78±0.02, parallax=0.0772±0.0054”, distance modulus=0.562±0.16) in comparison to the stars in the local volume, it looks like Gliese 1214 has of order twice solar metallicity if we adopt the nominal values for V,K and the distance. That’s very intriguing…

Georges-Louis Leclerc, Comte de Buffon is well known to givers of planet talks as one of the original proponents of physical cosmogony. Further fame accrues to his long-distance tangle with Thomas Jefferson over the size and the valor of the North American fauna. Buffon also made interesting contributions to probability theory, including the very sensible proposition that 1/10,000th is the smallest practical probability [source].

I think it’s reasonable to apply Buffon’s rule of thumb in discussing scenarios for the detection of the first potentially habitable extrasolar planet. If a scenario has a less than 10^-4 chance of unfolding, then it’s not worth expounding on in a web log post.

There’s no getting around the fact that the extrasolar planets are a long way away. Traveling at just under the speed of light, one reaches Alpha Cen Bb during Obama’s second term, and Gliese 581c, the extrasolar planet with the highest current value on the habitable planet valuation scale, lies 20 light years away. For practically-minded types such as myself, it’s depressing to think of the realistic prospects (or lack thereof) of actually reaching these worlds in a lifetime. And why spend trillions of dollars to visit Gliese 581 c when Venus is basically right next door?

It’s imperative to know the addresses of the nearest potentially habitable planets, though, and this is a goal that should be reached within roughly a decade or two. Barring a strike with some household name like Alpha Centauri or Tau Ceti, it’s a reasonable bet that the closest million-dollar world is orbiting a red dwarf.

The general suitability of red dwarf planets is often viewed with suspicion. Atmosphere-eroding flares, tidally spin-synchronized orbits, and gloomy formation-by-accretion scenarios provide ample material for space-age Jeremiahs. But first things first. With what frequency are Earth-sized T_eff~300K planets actually to be found in orbit around red dwarfs?

If planets form from analogs of the so-called Minimum Mass Solar Nebula, then the answer is quite well established: almost never.

If, however, instead of scaling down from the Minimum Mass Solar Nebula, we scale up from the proto-Jovian, proto-Saturnian and proto-Uranian disks, then the prospects are quite good. Ryan Montgomery and I have an Icarus preprint out which looks in detail at the consequences of an optimistic planet formation scenario for red dwarfs. Perhaps the most redeeming aspect of our theory is that it will be put to the test over the next decade. If hefty terrestrial planets are common around red dwarfs, then the currently operating ground-based MEarth survey will have an excellent chance of finding several examples of million-dollar wolds during the next several years, and the forthcoming TESS Mission will quite literally clean up.

In the spirit of Buffon, though, for the exact specifics of scenario three, it’s fun to probe right down to the limit of practical odds. Consider: An Earth orbiting a star at the bottom of the Main Sequence produces a transit depth that can approach 1%. If Barnard’s Star harbors an optimally sized and placed planet, then its value is a cool 400 million dollars. Such a planet would have an orbital period of about 13 days, and an a-priori transit probability of roughly 2%. I estimate a 1% chance that such a planet actually exists, which leads to a 1 in 5000 chance that it’s sitting there waiting for a skilled small-telescope observer to haul it in. In expectation, it’s worth $87,200, more than the equivalent of a Keck night, to monitor Barnard’s star at several milli-magnitude precision for a full-phase 13 days. That’s $280 dollars per hour. There are few better uses to which a high-quality amateur telescope could be put during those warm and clear early-summer nights.

Last weekend, I got e-mail from an A-list planet hunter who wrote in support of the little guys:

Why punish beloved M-dwarfs?

The last factor, currently written in terms of V, might be rewritten in terms of a less pejorative magnitude, like I or Z. Most stars in the Galaxy put their best (and brightest) foot forward at 1um!

Hard-working red dwarfs, like Barnard’s star or Proxima Centauri get the short end of the stick in the Oklo terrestrial planet valuation formula. Red dwarfs put out the bulk of their radiation in the near-infrared, rather than the optical, but dollar value is pegged to apparent magnitude in the V-band.

This leaves me in a position similar to that of a company spokesman trying to justify Wall Street bonuses.

“The fact of the matter, is that as a society, our planet-hunting values and priorities have been traditionally tied to the optical range of the spectrum. If we examine the resources that have been deployed to date, over a billion dollars have been spent on satellite-based planet-hunting programs that monitor stellar output in visible light. In the same way that an executive’s compensation is tied to the value that he or she brings to shareholders, a terrestrial planet’s value should therefore be tied to V-band magnitude.”

Flimsy, I admit. Therefore, in the interest of fairness, the first planet-hunting group or individual that discovers a planet worth USD 1M with Z-band apparent magnitude replacing V-band will receive an oklo.org T-shirt.